What is the best upper bound of the numbers of
-dimensional faces of a -polytope with vertices?

Let denote the number of -faces of a -polytope ,
for
.

The exact upper bound for in terms of and .
is known, thanks to McMullen's upper bound theorem.

The convex hull of distinct points on the moment curve
in
is known as a cyclic polytope. It is known that
its combinatorial structure (i.e. its face lattice, see
Section 2.3)
is uniquely determined by and .
Thus we often write to denote any such
cyclic -polytope with vertices.

McMullen's Upper Bound Theorem shows that the maximum
of is attained by the cyclic polytopes.